Example 47 - Chapter 6 Class 12 Application of Derivatives
Last updated at Jan. 7, 2020 by Teachoo
Transcript
Example 47 Find intervals in which the function given by f(๐ฅ) =3/10 ๐ฅ4 โ 4/5 ๐ฅ^3โ 3๐ฅ2 + 36/5 ๐ฅ + 11 is (a) strictly increasing (b) strictly decreasingf(๐ฅ) = 3/10 ๐ฅ4 โ 4/5 ๐ฅ^3โ 3๐ฅ2 + 36/5 ๐ฅ + 11 Finding fโ(๐) fโ(๐ฅ) = 3/10 ร 4๐ฅ^3 โ 4/5 ร 3๐ฅ^2 โ 3 ร 2x + 36/5 + 0 fโ(๐ฅ) = 12/10 ๐ฅ^3โ 12/5 ๐ฅ^2โ 6x + 36/5 fโ(๐ฅ) = 6/5 ๐ฅ^3โ 12/5 ๐ฅ^2โ 6x + 36/5 fโ(๐ฅ) = 6(๐ฅ^3/5โ(2๐ฅ^2)/5โ๐ฅ+6/5) fโ(๐ฅ) = 6((๐ฅ^3 โ 2๐ฅ^2โ 5๐ฅ + 6)/5) = 6/5 (๐ฅ^3โ2๐ฅ^2โ5๐ฅ+6) = 6/5 (๐ฅโ1)(๐ฅ2โ๐ฅโ6) = 6/5 (๐ฅโ1)(๐ฅ2โ3๐ฅ+2๐ฅโ6) = 6/5 (๐ฅโ1)[๐ฅ(๐ฅโ3)+2(๐ฅโ3)] = 6/5 (๐ฅโ1)(๐ฅ+2)(๐ฅโ3) Hence fโ(๐ฅ) = 6/5 (๐ฅโ1)(๐ฅ+2)(๐ฅโ3) fโ(๐ฅ) = 6(๐ฅ^3/5โ(2๐ฅ^2)/5โ๐ฅ+6/5) 6/5 (๐ฅโ1)(๐ฅ+2)(๐ฅโ3) = 0 (๐ฅโ1)(๐ฅ+2)(๐ฅโ3) = 0 Hence x = โ2 , 1 & 3 Plotting point on real line Thus, we get four disjoint intervals i.e. (โโ,โ2) ,(โ2, 1) ,(1 , 3), (3 , โ) โ f(๐ฅ) is strictly decreasing on the interval ๐ฅ โ(โโ,โ๐)& (๐ , ๐) f(๐ฅ) is strictly increasing on the interval ๐ฅ โ(โ๐,๐) & (๐ , โ)
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Chapter 6 Class 12 Application of Derivatives
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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.